Having offered and awarded a bounty on this question for effort more than results, after considerable analysis and effort I have come up with a solution - probably.

Given a number of this form $\sum\limits_{k=0}^{P-1}{b^k}$

Where b is an odd Prime
, P is Prime

the following primality test can be used to either confirm primality or highly probable primality (I have seen plenty of evidence to suggest the high probability but none to disprove the confirmation). Assume all variables are BigInteger (or equivalent)

initialize with res = $2^b$,
test = $\frac{b^P-1}{b-1}$

```
Repeat P-1 times
{
res = (res ^ b) Mod(test);
}
if after repeating (res = 2^b)
then test is Prime
```

So for the case where b = 3, and to answer the question:

Initialize with

res = 8 and test = $\frac{3^P-1}{2}$

```
Repeat P-1 times
{
res = (res ^ 3) MOD(test);
}
if after repeating (res = 8)
then test is Prime
```

I will explain how I came up with this and why I am not at all confident that it proves primality as the Lucas-Lehmer test has been shown to.

My initial aim was to produce an efficient Fermat Primality test for 2, as I had noticed that nearly all (if not all?) numbers of this form with b=3, that passed that test were primes.

So I set out to find an efficient way to calculate $2^N mod(N)$ where N is the sum of powers of 3.
Noticing that $2^{3^{n+1}} = {({2^{3^n}}})^3$ where n>0, and being familiar with the repeated squaring method, I recognized this as a candidate for repeated cubing.

However, when I tested this out I was surprised to find that the final phase of this method (i.e. multiplying all remainders mod(N)) was not necessary as the $P^{th}$ element was always 8. The reason for this is unclear to me, but maybe someone can answer this. I was also surprised to find that I could not get the test to fail when tested with a number of different prime bases and for thousands of prime exponents with base 3.

So the questions I would have about this would be:

- Is this just a shortcut way of doing a Fermat Primality test for witness 2?
- If it is, is there something special about numbers of this form with odd prime bases that means they are never Fermat Liars for witness 2?
- Is it just that there are very few such liars around and I, and probably others, have not yet exposed one?